In a small bowl, combine garlic with olive oil, lime juice, aji amarillo or hot sauce, ground cumin, paprika, oregano, salt, and pepper. Mix well until all ingredients are integrated into a smooth paste.
Next, prepare your chicken. Spatchcock it by removing the backbone to allow for even cooking. Gently loosen the skin around the breasts and thighs using your fingers, being careful not to tear it.
Using a small spoon, spread half of the marinade paste under the skin and the other half over the skin of the chicken. This ensures that every bite will be packed with flavor.
Place the chicken on a sheet pan and let it sit with the marinade for about 15 minutes while you preheat your oven or grill to 400°F (190°C).
If using the oven, roast the chicken for 30 minutes, checking on it occasionally and basting it with its own juices. After 30 minutes, cook for an additional 20 minutes or until the internal temperature reaches 165°F.
For grilling, oil your grill grates, then place the chicken breast side up over indirect heat. Cook for about 50 minutes, ensuring it also reaches that perfect 165°F.
While the chicken is cooking, prepare your cilantro sauce. In a food processor or blender, combine the cilantro, jalapeños, garlic, aji amarillo, olive oil, lime juice, vinegar, cotija cheese, and salt.
Blend all the ingredients until smooth, and then add mayonnaise, blending on low until just combined. Be cautious not to over-blend to maintain the emulsion of the mayo.
Allow the chicken to rest for 10 minutes after cooking before carving. This resting period ensures that all the juices are retained.
Serve the beautifully carved chicken with the cilantro sauce, accompanied by cilantro lime rice and lime wedges on the side.
